# Is there a, in depth, classification of branch points in complex analysis?

Disclaimer: This question was originally posted in math.stackexchange.com and, after 30 days with no answers, I followed the instructions of this topic.

In complex analysis we have well known results about isolated singularities. Poles are characterized by ‘nice’ (rational) controlled growth around them and for essential singularities we have the Great Picard's Theorem.

Question: Is there a similar classification for branch points? I mean: a clear list with all possibilities and results that characterizes each case?

For example: if we compare $$f(z)=\sqrt z$$ and $$g(z) = \sin(\ln(z))$$, they have very different behavior, one has a well defined limit as we approach $$z=0$$ in any branch and the other has an accumulation point of zeros. Are there results that characterize the ‘fast oscillations’ of $$g(z) = \sin(\ln(z))$$ and the ‘calm’ behavior of $$f(z)=\sqrt z$$? (Maybe in an appropriate Riemann surface.)

Yes, there is a classification. An isolated branch point can be algebraic or logarithmic. If the branch point is at 0, algebraic means that $$f(z^n)$$ has a pole or removable singularity at 0. It can also have an essential singularity, but this does not have an accepted name. In the case of a logarithmic point $$f(e^z)$$ is an (arbitrary) meromorphic function in some left half-plane. Anyway, plugging $$z^n$$ or $$e^z$$ (this is called local uniformization) reduces any classification at an isolated branch point to the case of a single valued function.

Literature: on singularities of analytic functions in general, and their classification, there is a book

P. Dienes, Leçons sur les singularités des fonctions analytiques; professées à l’université de Budapest, Paris: Gauthier-Villars, VIII + 172 S. 8∘ (1913),

which is somewhat out of date.

"Isolated branch points" by themselves are not a subject of a special study, but "functions whose all singularities are isolated and may include isolated branch points" play an important role in Ecalle's theory of "resurgent functions". Jean Ecalle wrote several books on them, but they are very hard to read. Of the many expositions of Ecalle's theory I can mention

C. Mitschi and D. Sauzin, Divergent series, summability and resurgence I. Monodromy and resurgence. Lecture Notes in Mathematics 2153, Springer (ISBN 978-3-319-28735-5/pbk; 978-3-319-28736-2/ebook). xxi, 298 p. (2016).

• So $f(z)=\sqrt z$ has an algebraic branch point as $f(z^2)=z$ has a removable singularity at $0$. $g(z) = \sin(\ln(z))$ has an logarithmic branch point as $g(e^z) = \sin(z)$ is meromorphic in $Re(z)<0$. (I don't get this part, should not be in a neighborhood of $0$?) But $h(z)=\sin(\frac{1}{z^{1/3}})$ has no accepted name as $h(z^3)=\sin(\frac{1}{z})$ has an essential singularity. Right? What about results characterizing then? Could you suggest any reference about this topic? I'm doing my doctorate in physics, so something more "example/intuition" oriented is prefered over "proof" oriented. Nov 11, 2021 at 13:44
• Your examples are correct. $e^z$ maps any left half-plane onto a punctured neighborhood of $0$. So $f$ initially defined in a punctured neighborhood of $0$ becomes defined in a left half-plane. I do not understand what references you are asking for. For terminology? The terminology is not so well established as for isolated singularities of single valued functions. For theorems about such isolated singularities? There are many theorems, like there are many theorems on isolated singularities of single valued functions. So be more specific. Nov 11, 2021 at 14:34